Maddenation

The Puzzle Instinct

I have been reading the book The Puzzle Instinct, by Marcel Danesi, which was a birthday gift from my childhood friend, Kaz. He knew that, no matter how many puzzle books I have or have read, that another one would still be welcome. Here are some tidbits.

I learned that Fibonacci’s real name was Leonardo Pisano, and that Fibonacci is an acronym for figlio Bonacci, “son of the Bonacci family.” (He also sometimes called himself Bigollo, which means “bungling traveler.” The Fibonacci series originated as the solution to his “Rabbit Puzzle,” which asked how many pairs of rabbits are produced from a single pair if each pair reproduces every month and new pairs are fertile from the second month on. The monthly totals form the Fibonacci series we all know so well. (1,1,2,3,5,8,13…)

One new Fibonacci fact I learned from Danesi is that, if the nth number in the sequence is x, then every nth number after x is a multiple of x! For example, the 3rd number is 2, and every third number after 2 is a multiple of 2 (8, 34, 144,…). Check it our for other numbers.

Even more amazing is a surprising relationship to magic squares. Magic squares are arrangements of numbers in a square pattern in which all the rows, columns, and diagonals add up to the same number. The simplest is the 3×3 pattern shown below:

8 1 6
3 5 7
4 9 2

If one substitutes for these numbers the corresponding Fibonacci number, a new “magic square” is produced in which the sum of the products of of the three rows is equal to the sum of the products of the three columns! OK, well, maybe that’s not too amazing. But you can check it out for yourself.

I can’t quit before I present one more problem from the book. This one is a favorite of mine that dates back to the fifteenth century and was first penned by Nicolas Chuquet.

You have two jars that will hold 5 and 3 pints respectively, neither jar being marked in any way. How can you measure exactly 4 pints from a cask with an unspecified quantity of liquid in it, given that you are allowed to pour liquid back into the cask?

Comments

Oh, come on, Dad! Haven’t you seen Die Hard III: With a Vengeance? (Bruce Willis and Samuel L. Jackson have to do this problem in order to keep a briefcase from exploding in a NYC park.)

Fill the 3pint. Dump it into the 5pint. Fill the 3pint again. Dump 2 of the 3pint into the 5pint. You have 1 in the 3pint now. Empty the 5pint. Pour the 1 from the 3pint into the 5pint. Fill the 3pint. Dump it into the 5pint (already containing 1).